# Is there a lower bound on the average diamond norm of two uniformly random unitaries U1 and U1 of dimension D that are sampled from haar measure?

I could not find any lower bound on the diamond norm for two uniformly random unitaries of dimension D sampled from the haar measure.

First, suppose that $$\Phi_0(X) = U_0 X U_0^{\dagger}$$ and $$\Phi_1(X) = U_1 X U_1^{\dagger}$$ are any two unitary channels. In this special case, the diamond norm distance between these two channels can be expressed as follows: $$\|\Phi_0 - \Phi_1\|_{\diamond} = 2 \sqrt{1 - \delta(U_0^{\dagger} U_1)^2},$$ where $$\delta(A)$$ denotes the minimum absolute value taken over the numerical range of a given operator $$A$$. That is, $$\delta(A) = \min_{|\psi\rangle} |\langle \psi | A | \psi\rangle|,$$ where the minimum is over all unit vectors $$|\psi\rangle$$.
Now, if you're interested in $$U_0$$ and $$U_1$$ being Haar-random, then you might as well set $$U_0 = I$$ and $$U_1 = U$$ for $$U$$ Haar-random, by the fact that the diamond norm is unitarily invariant. To understand the distribution of the diamond norm distance, you therefore need to understand something about the numerical range of a Haar-random unitary. Because unitary operators are normal, their numerical range is the convex hull of their eigenvalues, so you're essentially trying to understand something about the eigenvalues of a Haar-random unitary. This topic has been studied extensively in random matrix theory.
I do not know enough about this topic to give you precise bounds, but it is clear that as the dimension $$D$$ grows, the probability that the diamond norm distance equals the maximum possible value 2 (which means the two channels can be discriminated perfectly) approaches 1. Indeed, in order to have diamond norm distance strictly less than 2, all of the eigenvalues of $$U$$ (or $$U_0^{\dagger} U_1$$) must fall within an arc on the unit circle having length strictly less than $$\pi$$, which is extremely unlikely for large $$D$$ (surely the probability is exponentially small in $$D$$).